For hypergraphs, analogues of results of Sudakov–Vu [1]} were proven by Ferber and Kwan [2]} for \(k\) -uniform hypergraphs where \(p\gtrsim \max \lbrace n^{-k/2+o(1)},n^{-k+2}\log n\rbrace \) . In particular, for \(G\sim \mathbb {G}^{(k)}(n,p)\)  with \(p\)  above the specified threshold, with high probability any \(G^{\prime }\subseteq G\)  satisfying \(\delta _{\ell }(G^{\prime })\ge {(\delta _{\ell ,k}^+ + \varepsilon )\binom{n-\ell }{k-\ell }p}\)  contains a perfect matching. (The case when \(\ell = k-1\)  was handled in earlier work of Ferber and Hirschfeld [3]}.) Note that this result is nontrivial only when \(p = \Omega (n^{\ell -k}\log n)\)  since otherwise one has \(\delta _{\ell }(G) = 0\)  and there are no subgraphs satisfying the hypothesis of the statement. This indicates a difference between the notion of robustness considered in our work and the notion of local resilence considered in these previous works as the probabilistic ranges of interest in thm:pm and [2]} only coincide when \(\ell = 1\) .
